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Study in the app →Mathematics · Year 12 (Methods 3–4) · Chapter 1
The slope at a single point
Practice
- d/dx of x²
- 2x
- d/dx of x³
- 3x²
- d/dx of x⁵
- 5x⁴
- d/dx of 5x
- 5
- d/dx of 7
- 0 — constants don't change
- d/dx of 3x²
- 6x
- d/dx of x² + 5x
- 2x + 5
- slope of y = x² at x = 3
- 6 — evaluate 2x there
- the derivative measures
- the instantaneous rate of change — the slope at a single point
- average vs instantaneous rate
- whole-interval slope vs the slope right now
Easy questions
- d/dx of x²
- 2x
- d/dx of x³
- 3x²
- d/dx of 4x
- 4
- d/dx of 9
- 0
- d/dx of x⁵
- 5x⁴
Medium questions
- d/dx of 3x²
- 6x
- d/dx of x² + 5x
- 2x + 5
- slope of y = x² at x = 3
- 6
- d/dx of x³ − x
- 3x² − 1
- d/dx of 2x⁴
- 8x³
Hard questions
- slope of y = x³ at x = −1
- 3 — 3x² is 3(1)
- Where is y = x² flat?
- x = 0 — solve 2x = 0
- y = x² − 4x: where is the slope zero?
- x = 2 — the bottom of the parabola
- Why does the +3 in x² + 3 vanish when differentiating?
- Constants have zero rate of change — they lift the curve, not its slope
- Average rate of y = x² from x = 1 to x = 3 vs instantaneous at x = 2
- Both are 4 — (9−1)/(3−1) = 4 and 2(2) = 4. The average over an interval is hit exactly somewhere inside it.
Lesson
Zooming in on a curve
Average speed is a whole-trip fact: distance over time. But zoom in on any smooth curve far enough and it straightens into a line — and THAT line's slope is your rate at that exact instant. That's the derivative: the slope of a curve at a single point, the speed the camera saw. For powers of x there's a clean pattern: bring the power down front, drop it by one. x² → 2x, x³ → 3x², x⁵ → 5x⁴. Constants like 7 don't change at all, so their derivative is 0, and a coefficient just rides along: 3x² → 6x. The derivative is itself a formula — 2x is the slope of x² EVERYWHERE. Want it at x = 3? Evaluate: slope 6. Your average over the hour and your speed at the camera are different questions — calculus is the mathematics of the second one.
- d/dx x² = 2x — at x = 3 the curve climbs at slope 6.
- d/dx x³ = 3x²; d/dx x⁵ = 5x⁴ — power down front, drop by one.
- d/dx 7 = 0 — a flat line has no slope anywhere.
- d/dx (x² + 5x) = 2x + 5 — differentiate term by term.
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